Harvard University interview exponential math question|Can you pass this exam?

hi let's solve for the value of x okay so we are told that x squared minus x plus one is equal to zero if this is equal to zero, what is x to the power of 20 plus x to the power of 10 10 minus one, okay. Now the first thing we need to do, let's concentrate on the first equation that says that x squared minus x plus one equals zero; now I want to make x squared the subject of the formula, so I will have that x squared is equal to minus x; we move it to become plus x And plus one becomes minus one. Now we are going to call this Equation One. All right. Now from here, from this point again, we can say that x squared minus x is equal to what? We cross to become minus one. Now I want to factorize this. Okay, so if I bring out x from here. x squared divided by x gives us x; minus x divided by x gives us one; is equal to minus One, all right, now let's divide both sides by x, divide by x, so we'll have that x minus one, right? So we have that x minus one is also equal to minus one over x. Now let's call this equation. two, all right. Equation two. Now, if we compare equation one and equation two, we can still say If x squared equals x minus one, that simply means that x squared also equals minus one. over x since it is equal to x minus one, right? So we can write that x squared is equal to minus one over x, so we can call this equation three. NowPropertyChanged x because I want to raise x to the power of three before I start simplifying what we have here All these things I'm bringing out here are the ones we will need when we are simplifying. this one is okay, all right, so we have x squared multiplied by x is equal to minus one over x multiplied by x over one okay all right now when you have a to the power of m multiplied by a to the power of n is equal to a to the power of m plus what? n, okay, so that means that We can write this as what times x squared plus one, because this is two and this is one. x cubed is equal to minus one, because this is what we can see here. Now this is another important equation; let's call it Equation Four. Okay, if this is the first time Across this channel, please subscribe to the channel and turn on your notification bell. so I'll be notified anytime I upload a new video, and please give it a thumbs up if you You're actually enjoying the video, thank you. All right, now let's go straight to simplifying this. I'm going to start with x to the power of 2020; I will simplify it first, okay? Now I want us to note that x to the power of 2020 can be written as x to the power of one times x to the power of 2019, so this implies that this is equal to or let's write this as equal to x to the power of one multiplied by x to the power of three bracket because I want to divide this by three, okay, since I have x to the power of three, recall that we have x to the power of three here, okay? All right, now 20 divided by 3 will give us 60. The remainder is 2 when we divide 21; we get 7. When we divide 9, we get 3. That's what we have. So this simply means that x to the power of 20 is x multiplied by x to the power of 3, recall. Is what minus 1 all right? So we have minus 1 to the power of 673. Now this is equal to what x? multiplied by recall that minus 1 since this is odd minus 1 to the power of any odd number still give us minus 1, so multiplied by minus 1, so this simply implies that x to the power of 2020 is equal to minus x; we are going to keep this important information or equation. Now let's analyze the second one, the second one which is x to the power of 10, so we'll write x to the 10 to the power of 10 is the same as x to the power of 2. Why am I bringing this up if I bring out 3? This cannot be divided by 3, okay, so I'm bringing it out because I know I have x to the power of 2. Somewhere I know I have it, somewhere I can easily substitute so x to the power of 2 multiplied by x to the power of when 11010 minus 2, you're going to have 1008, right? 1008. Now this can be divided by 3 as well, right? So we now have this is equal to x to the power of 2 multiplied by x cubed times 10 divided by 3 gives us 3 with a remainder of 1; if we divide another 10 We have 3 with a remainder of 1 when we divide 18; we have 6, okay, so that means this is equal to x to 2 raised to the power of x cubed equals minus 1, okay? Recall, so we have minus 1 raised Now, because 336 is even, minus 1 to the power of 336 gives us plus 1, right? So we are going to have plus 1, so this simply means that x to the power of 10 plus 10 is equal to x squared, all right. because x squared multiplied by 1 still gives us x squared, now this is another important Information now, it is time for us to do it to simplify. So this is where we are right now—we are I'm going to replace all these things we have in here, all right, and I don't want to clean anyone. So I'm going to write down the important ones: x squared is equal to x minus 1, and I think then x squared minus x equals minus 1, so these are the important ones there before I clean here Alright, so let me go ahead and clean up now and record that x to the power of 2020 after simplification. We got it to be minus x, so I'll write minus x plus now x to the power of 10, 10 x to the power of 10 10 gave us what x squared, so we have x squared, all right, minus 1, and this is equal to recall that x squared minus x is part of what I cleaned; x squared minus x gives us minus 1, all right, so I wrote it somewhere, but x squared, which is this, minus x is equal to minus 1, so we are going to substitute it here. This implies that this plus this All right, that is x squared minus x equals minus 1, and we have minus 1 here, and this is equal. to minus 2, so this simply means that x to the power of 2020 plus x to the power of 1010 minus I will give us minus 2 if x squared minus x plus 1 equals zero.